In previous Annual Reports we described our approach to summarizing the shape of neuronal dendrites. Measurements from 64 or more reconstructed motoneuron dendrites were interpreted as the result of a hypothetical probabilistic branching process. The probabilities were measured and became parameters for a stochastic algorithm which produced sample dendrites having the statistical properties of observed dendrites. In this report, we describe new methods to predict the average properties of the complete ensemble of these sample dendrites, given the parameters used by the program that produces them. All our methods resemble the stochastic model because they take the configuration of branch diameters at one distance from the stem and transform it into that at the next distance increment. Last year we described a matrix H embodying the model probabilities which generates the probability distributions for branch diameter at all distances. This year this generator has been reduced to a differential equation, so that the properties of our original model and of the structure of dendrites is much clearer. We also have found that the method of generating functions can be used to directly generate the probabilities of complex combinations of branch diameter. This should help us find what properties of motoneurons, which may be complex functions of branch diameter distributions, reach a maximum for the observed values of their shape parameters. Also we have contributed to an operational definition of "Lacunarity", a fractal measure of shape used by Dr. T. G. Smith to characterize cultured neurons.